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NUMBER SYSTEMS (Integers INTEGERS (Non-negative integers Non-negative…
NUMBER SYSTEMS
INTEGERS
Non-negative integers (natural numbers)
Positive Integers
Identity element
Multiplicative
(m)(1) = m
Properties
Associativity
(m+n)+k = m+(n+k)
(mn)k = m(nk)
Commutativity
m+n = n+m
mn = nm
Distributivity
m(n+k) = mn + mk
Prime numbers
2, 3, 5, 7, 11, ...
n Factorial
n! = (n)(n-1)(n-2)...(3)(2)(1)
Number zero
Multiplication by zero
(m)(0) = 0
Absence of zero divisors
If mn = 0 then m = 0 or n = 0
Properties
Linearity
m > n or m < n or m = n
Additive inverse
m + (-m) = 0 eg. 4 + (-4) = 0
Transitivity of = and <
If m = n and n = k then m = k
If m < n and if n < k then m < k :
Monocity of + and x
Plus: If m = n then m+k = n+k and mk = nk
Multiplication: If m < n then m+k < n+k and if k > 0 then mk < nk
Identity element
Multiplicative (1)
(m)(1) = m
Additive (0)
m+0 = m
Monocity
If m = n then m+k = n+k and mk = nk
If m < n then m+k < n+k
If m < n and k > 0 then mk < nk
If m < n and k < 0 then mk > nk
Multiplication rules
(-)(+) = -
(-)(-) = +
(+)(-) = -
(+)(+) = +
Absolute value
|m| = m if m is non-negative e.g. |2| = 2
|m| = -m if m is negative e.g. |-2| = 2
Additive inverse
For every m there exists an integer n such that m+n = 0
e.g. 4 + (-4) = 0
e.g. -4 + (4) = 0
RATIONAL NUMBERS
IRRATIONAL NUMBERS
